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Mirrors > Home > MPE Home > Th. List > elndif | Structured version Visualization version GIF version |
Description: A set does not belong to a class excluding it. (Contributed by NM, 27-Jun-1994.) |
Ref | Expression |
---|---|
elndif | ⊢ (𝐴 ∈ 𝐵 → ¬ 𝐴 ∈ (𝐶 ∖ 𝐵)) |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | eldifn 4101 | . 2 ⊢ (𝐴 ∈ (𝐶 ∖ 𝐵) → ¬ 𝐴 ∈ 𝐵) | |
2 | 1 | con2i 141 | 1 ⊢ (𝐴 ∈ 𝐵 → ¬ 𝐴 ∈ (𝐶 ∖ 𝐵)) |
Colors of variables: wff setvar class |
Syntax hints: ¬ wn 3 → wi 4 ∈ wcel 2105 ∖ cdif 3930 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1787 ax-4 1801 ax-5 1902 ax-6 1961 ax-7 2006 ax-8 2107 ax-9 2115 ax-10 2136 ax-11 2151 ax-12 2167 ax-ext 2790 |
This theorem depends on definitions: df-bi 208 df-an 397 df-or 842 df-tru 1531 df-ex 1772 df-nf 1776 df-sb 2061 df-clab 2797 df-cleq 2811 df-clel 2890 df-nfc 2960 df-v 3494 df-dif 3936 |
This theorem is referenced by: peano5 7594 extmptsuppeq 7843 undifixp 8486 ssfin4 9720 isf32lem3 9765 isf34lem4 9787 xrinfmss 12691 restntr 21718 cmpcld 21938 reconnlem2 23362 lebnumlem1 23492 i1fd 24209 hgt750lemd 31818 fmlasucdisj 32543 dfon2lem6 32930 onsucconni 33682 meaiininclem 42645 caragendifcl 42673 |
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